preview

Binomial Model : Black Scholes

Good Essays
Open Document

Rajdeep Atwal
December 2015
The University of Leicester

CT8: Financial Mathematics

Black Scholes and Binomial Model

Contents

Literature Review 3
Binomial model 3
Black Scholes 5
Justification 7
Binomial model 7
Black Scholes 11
Modified Exam Question 11
Binomial model 13
Black Scholes 16
Bibliography 16

Literature Review
These models are very similar, they are based on similar theoretical foundations and have a series of assumptions, such as; Geometric Brownian Motion and risk-neutral valuation. However, both models have unique differences which are foundational in there model, some of which are highlighted below.
Binomial model A binomial model in a discrete time provides a method for evaluating options. Contracts …show more content…

[1]
In the Cox and Rubinstein method explained above, once n →∞ you gain approximations for your mean and variance, this suggests given a period of time the mean and variance of this model will approximately be a close match in a risk neutral world. Tian suggested to match the first three moments of the binomial model to the first three moments of a lognormal distribution. This is using the mean, variance and skewness being the first raw moment, second central moment and third central moment respectively. The Tian tree could be parameterized so that it could be interpreted more fundamentally as a binomial tree in which every other period were simply skipped. Which leads to three equations, which when we solve we gain the following parameters. pu^3+(1-p) d^3=(〖e^(r△t))〗^3 (〖e^(σ^2△t))〗^3 pu^2+(1-p) d^2=(〖e^(r△t))〗^2 e^(σ^2△t) u=〖0.5e〗^(r△t) υ(υ+1+√(υ^2+2υ-3)),d=〖0.5e〗^(r△t) υ(υ+1-√(υ^2+2υ-3)) p= (e^(r△t)-d)/(u-d),υ=e^(σ^2△t)
Leisen and Reimer decided to try and establish and method which improves the binomial tree model. They developed a model by improving the rate which the binomial tree converges to. In real terms, they suggested to change probabilities of an up move in both stock and bond measures which leads to the up and down moves. Leisen-Reimer tree is calculated using the following parameters:
Where h^-1 is a discrete approximation of the c.d.f for a normal

Get Access
Related
Get Access